Bispaces admitting only bicomplete or only totally bounded quasi-metrics

Abstract

We characterize quasi-metrizable bispaces that admit only bicomplete quasimetrics by means of doubly primitive sequences, and deduce that if (X, S, T) is a quasi-metrizable bispace admitting only bicomplete quasi-metrics and either (X, S) or (X, T) is hereditarily Lindelöf, then (X, S ∨ T) is compact. We also give an example which shows that hereditary Lindelöfness cannot be omitted in the above result. Finally, we show that a quasi-pseudometrizable bispace (X, S, T) admits only totally bounded quasi-pseudometrics if and only if (X, S ∨ T) is compact, and deduce that a quasi-pseudometrizable topological space admits only totally bounded quasi-pseudometrics if and only if it is hereditarily compact and quasi-sober (equivalently, if and only if it admits a unique quasi-uniformity).

AMS (1991) Subject classification

54E35 54E55 54D30 54E15

Key words and phrases

The first author acknowledges support from the Swiss national Science Foundation, under grant 2000-056811.99 and from the Universidad Politécnica de Valencia. The second author acknowledges the support of the DGES, under grant PB95-0737. The third author acknowledges the support of the Universidad Politécnica de Valencia, as well as the provision of leave by the University of South Africa.